Rory,
I'm not entirely clear on the layout you're describing, but if the two woofers are in sealed cabinets, and both are open to the room through the same general opening, then they will not cancel, assuming that they are wired in phase with each other.
If you're using vented cabinets, all bets are off, as the output will depend on the positioning of the vents relative to the drivers and the frequency that you're reproducing at the moment.
Note that if you're facing two woofers towards each other, they will load each other by increasing/decreasing the pressure on the other driver.
What are you hoping to accomplish with this arrangement?
Whether to wire drivers in series or parallel is an ugly topic, open to a lot of interpretation. Wiring woofers in series decreases the control the amplifier has over the driver (the damping factor) by adding the impedance of the other driver to the Zout of the amplifier. If you're using low impedance drivers, it may be necessary in order to avoid creating a load that is too low for your amplifier to drive comfortably. In most cases, it is better to wire drivers in parallel.

I intended to use this design for an omnidirectional sound, which would radiate throughout 360 degrees. The woofers are in separate ported boxes and while the woofers face each other, the ports face forward, the way that the tweeters face.

Rory,
Depending on frequency, driver diameter, etc. low frequencies are already pretty much omni-directional. Is this a two-way, or a three way?
Also, keep in mind that omni-directional speakers can be a real nuisance to place in a room. (*Strong* room reflections.)
If omni is what floats your boat, you might want to dredge around and find info on the old Ohm speakers (are they still in business?), using a variation of the Walsh driver (I believe there were also Walsh speakers at one point). The essential concept being that the driver is facing down (place your woofer face down on a cardboard box to help visualize) into the enclosure. The back side of the driver (magnet & all) is open to the room on all sides, and is thus omnidirectional. They were capable of some really spooky imaging effects, but that was pretty much their only claim to fame. Something like that might give you a better approach. Incidentally, the Ohm drivers were ported, as I recall, at the bottom of the cabinet, but there's no law that says you have to port them at all. If you already have two woofers on hand, it shouldn't be too hard to contrive a dual cabinet where one woofer fires down into a cabinet, the other--on some healthy standoffs--fires up into a separate cabinet. (Just make two identical cabinets, and turn one upside-down.) With the two drivers wired in phase. Mount your tweeters in the same are where the woofers are open. This would give you something of a D'Appolito MTM arrangement.

What you propose is certainly workable and it's one possible solution to not having to look at the woofers (more beautiful wood to view). Rear firing woofers as well as a bandpass box are another way to go. As GRollins says, the low frequencies are omni-directional to begin with so your proposition is not necessary to have this omni-directionally. Make sure they are symmetrical, vent faces vent, woofer faces woofer. Vents facing any other way might be OK as well. And of course the cabinets MUST be joined relative to each other. You will be de-tuning what you might have calculated as a proper low frequency system but you can play with this and restore proper operation. Sorry, I don't have formulae for this. Needless to say?, the woofers must be wired in phase.

This is NOT a workable solution for a two way speaker. The volume enclosed by the spacing between the boxes and perimeter of the boxes forms a resonant cavity. The resonant frequency of this cavity is going to be in the mid-range area; you must crossover to a mid-range well before this point.

I must advise that the woofers face each other horizontally. If you have one woofer facing down and one facing up, the cones will sag over time. This problem can be avoided if you design the bass boxes so they can be flipped over from time to time.

But, after all this, why bother? You can have what you want via less painful and time consuming routes.